A151310 Number of walks within N^2 (the first quadrant of Z^2) starting at (0,0) and consisting of n steps taken from {(-1, -1), (-1, 1), (-1, 0), (0, 1), (1, -1), (1, 1)}.

1, 2, 9, 37, 181, 869, 4430, 22640, 118808, 627275, 3358307, 18087833, 98248087, 536372711, 2945032779, 16237200915, 89896571332, 499383074373, 2783038880080, 15552412965005, 87134774936870, 489297438647055, 2753406974342080, 15523739339340106, 87677875795392958, 496006562740402954

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..250

Alin Bostan and Manuel Kauers, Automatic Classification of Restricted Lattice Walks, arXiv:0811.2899 [math.CO], 2008.

Mireille Bousquet-Mélou and Marni Mishna, Walks with small steps in the quarter plane, arXiv:0810.4387 [math.CO], 2008.

Marni Mishna and Juan Pulido, On the small-step quarter plane lattice walks with a non D-finite univariate generating function, arXiv:2605.16688 [math.CO], 2026. See p. 13 (Table 5).

Maple

steps:= [[-1, -1], [-1, 1], [-1, 0], [0, 1], [1, -1], [1, 1]]:
f:= proc(n, p) option remember; local t;
  if n <= min(p) then return 6^n fi;
  add(procname(n-1, t), t=remove(has, map(`+`, steps, p), -1));
end proc:
map(f, [$0..100], [0, 0]); # Robert Israel, Jun 11 2019

Mathematica

aux[i_Integer, j_Integer, n_Integer] := Which[Min[i, j, n] < 0 || Max[i, j] > n, 0, n == 0, KroneckerDelta[i, j, n], True, aux[i, j, n] = aux[-1 + i, -1 + j, -1 + n] + aux[-1 + i, 1 + j, -1 + n] + aux[i, -1 + j, -1 + n] + aux[1 + i, -1 + j, -1 + n] + aux[1 + i, j, -1 + n] + aux[1 + i, 1 + j, -1 + n]]; Table[Sum[aux[i, j, n], {i, 0, n}, {j, 0, n}], {n, 0, 25}]

Crossrefs

Sequence in context: A150990 A150991 A150992 * A000663 A364088 A218942

Adjacent sequences: A151307 A151308 A151309 * A151311 A151312 A151313

Keyword

nonn,walk

Author

Manuel Kauers, Nov 18 2008

Status

approved